Monday, 22 September 2014

On January 26th-27th 2015, the Faculty of Philosophy of the University of Groningen will host a short Winter School aimed at advanced undergraduate students and early-stage graduate students. The theme of the winter school is Paradoxes and Dilemmas, and it will consist of 6 tutorials where the topic will be discussed from different viewpoints: theoretical philosophy, practical philosophy, and the history of philosophy. As such, the Winter School may be of interest to at least some of the M-Phi readers; for further details, check the site of the Winter School.

Lectures:

Catarina Dutilh Novaes: ‘Paradoxes: at the heart of philosophy’

Barteld Kooi: ‘Epistemic paradoxes: is the concept of knowledge incoherent?’

Scholarships:
The Faculty is offering up to three EUR 300 scholarships for the best students enrolling in the winter school, and who express serious interest in later applying for the Research Masters’ program. Moreover, participants who are then accepted in the Research Masters’ program for the year 2015/2016 will have their registration fee for the winter school reimbursed.
To apply for the scholarships, send a short CV (max 2 pages) and a letter (max 1 page) stating your interest in the Faculty of Philosophy in Groningen and the Research Masters’ program in particular, to winterschoolphilosophy 'at' rug.nl with 'Application for winter school scholarship' as subject. Deadline to apply for the scholarships: December 1st 2014. Preference will be given to members of underrepresented groups in philosophy (women, people of color, persons with disabilities etc.).

Registration:
To register, send an email with your name, affiliation and status (undergraduate, graduate) to winterschoolphilosophy 'at' rug.nl with 'Registration for winter school' as subject, no later than December 15th 2014. As the number of spots is limited, you are encouraged to register early.

Friday, 19 September 2014

I was asked to write a review of Terry Parsons' Articulating Medieval Logic for the Australasian Journal of Philosophy. This is what I've come up with so far. Comments welcome!
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Scholars working on (Latin) medieval logic can be viewed as populating a spectrum. At one extremity are those who adopt a purely historical
and textual approach to the material: they are the ones who produce the
invaluable modern editions of important texts, without which the field would to
a great extent simply not exist; they also typically seek to place the
doctrines presented in the texts in a broader historical context. At the other
extremity are those who study the medieval theories first and foremost from the
point of view of modern philosophical and logical concerns; various techniques
of formalization are then employed to ‘translate’ the medieval theories into
something more intelligible to the modern non-historian philosopher. Between
the two extremes one encounters a variety of positions. (Notice that one and
the same scholar can at times wear the historian’s hat, and at other times
the systematic philosopher’s hat.) For those adopting one of the many
intermediary positions, life can be hard at times: when trying to combine
the two paradigms, these scholars sometimes end up displeasing everyone (speaking
from personal experience).

Terence Parsons’ Articulating
Medieval Logic occupies one of these intermediate positions, but very close to the second extremity; indeed, it represents
the daring attempt to combine the author’s expertise in natural language
semantics, linguistics, and modern philosophy with his interest in medieval
logical theories (which arose in particular from his decade-long collaboration
with Calvin Normore, to whom the book is dedicated). For scholars of Latin medieval
logic, the fact that such a distinguished expert in contemporary philosophy and
linguistics became interested in these medieval theories only confirms what
we’ve known all along: medieval logical theories have intrinsic systematic
interest; they are not only curious museum pieces.

Despite not being the first to employ modern logical
techniques to analyze medieval theories, Parsons' approach is quite unique (one
might even say idiosyncratic). It seems fair to say that nobody has ever before attempted to achieve what he wants to achieve with this book. A passage from
the book’s Introduction is quite revealing with respect to its goals:

Tuesday, 16 September 2014

In December, I will be presenting at the Aesthetics in Mathematics conference in Norwich. The title of my talk is Beauty, explanation, and persuasion in mathematical proofs, and to be honest at this point there is not much more to it than the title… However, the idea I will try to develop is that many, perhaps even most, of the features we associate with beauty in mathematical proofs can be subsumed to the ideal of explanatory persuasion, which I take to be the essence of mathematical proofs.

As some readers may recall, in my current research I adopt a dialogical perspective to raise a functionalist question: what is the point of mathematical proofs? Why do we bother formulating mathematical proofs at all? The general hypothesis is that most of the defining criteria for what counts as a mathematical proof – and in particular, a good mathematical proof – can be explained in terms of the (presumed) ultimate function of a mathematical proof, namely that of convincing an interlocutor that the conclusion of the proof is true (given the truth of the premises) by showing why that is the case. (See also this recent edited volume on argumentation in mathematics.) Thus, a proof seeks not only to force the interlocutor to grant the conclusion if she has granted the premises; it seeks also to reveal something about the mathematical concepts involved to the interlocutor so that she also apprehends what makes the conclusion true – its causes, as it were. On this conception of proof, beauty may well play an important role, but its role will be subsumed to the ideal of explanatory persuasion.

There is a small but very interesting literature on the aesthetics of mathematical proof – see for example this 2005 paper by my former colleague James McAllister, and a more recent paper on Kant’s conception of beauty in mathematics applied to proof by Angela Breitenbach, one of the organizers of the meeting in Norwich. (If readers have additional literature suggestions, please share them in comments.) But perhaps the locus classicus for the discussion of what makes a mathematical proof beautiful is G. H. Hardy’s splendid A Mathematician’s Apology (a text that is itself very beautiful!). In it, Hardy identifies and discusses a number of features that should be present for a proof to be considered beautiful: seriousness, generality, depth, unexpectedness, inevitability, and economy. And so, one way for me to test my dialogical hypothesis would be to see whether it is possible to provide a dialogical rationale for each of these features that Hardy discusses. My prediction is that most of them can receive compelling dialogical explanations, but that there will be a residue of properties related to beauty in a mathematical proof that cannot be reduced to the ideal of explanatory persuasion. (What this residue will be I do not yet know).

As I mentioned, this is still very much work in progress, but for now I would like to sketch what a dialogical account of beauty in a mathematical demonstration might look like for a specific feature. Now, a fascinating desideratum for a mathematical proof, which has been discussed in detail recently by Detlefsen and Arana, is the ideal of purity:

Throughout history, mathematicians have expressed preference for solutions to problems that avoid introducing concepts that are in one sense or another “foreign” or “alien” to the problem under investigation. (Detlefsen & Arana 2011, 1)

A mathematical proof is said to be pure if it does not rely on concepts that are not present in the statement of the conclusion of the proof (the theorem). Many famous mathematical proofs are not pure in this sense, such as Wiles’ proof of Fermat’s Last Theorem, which utilizes incredibly sophisticated and complex mathematical machinery to prove a theorem the statement of which can be understood with knowledge of standard high school level mathematics. (The impurity of Wiles’ proof is one of the motivations often given to seek for alternative proofs of FLT, as described in this guest post by Colin McLarty.) Now, I take it to be fairly obvious that purity concerns can be readily understood as aesthetic concerns, in particular related to simplicity (which is one of the features widely associated with beauty).

What would a dialogical account of the purity desideratum look like? Going back to the idea that the function of a proof is that of eliciting persuasion by means of understanding in an interlocutor (hence the stress on the explanatory dimension), it is clear that, in general, the less complex the mathematical machinery of a proof, the less it will demand of the interlocutor being persuaded in terms of cognitive investment. Moreover, if it relies on simpler machinery, the proof will most likely reach a larger audience, i.e. be persuasive for a larger number of people (those possessing mastery of the concepts used in it). In particular, a proof that only uses concepts already contained in the formulation of the theorem will be at least in theory comprehensible to anyone who can understand the statement of the conclusion. Thus, a pure proof maximizes its penetration among potential audiences, as it only excludes those who do not even grasp the statement of the theorem in the first place. In other words, purity sets the lower bound of cognitive sophistication required from an interlocutor precisely at the right place. (Naturally, I can also be convinced of the truth of a theorem even if I do not understand the proof myself, i.e. by relying on the expertise of the mathematical community as a whole.)

As I said, these are only tentative ideas at this point, so I look forward to feedback from readers. In particular, I would like to hear from practicing mathematicians their answers to the question in the title: what makes a mathematical proof beautiful? Do you agree with Hardy's list? (I could definitely use some input so as to render my investigation more in sync with actual practices!)

Wednesday, 10 September 2014

On behalf of the M-Phi contributors, I want to sincerely apologize to our readers for the misguided and inappropriate post that was online at M-Phi for four days (now taken down, as well as all other posts referencing the Oxford events). The moderation structure of the blog was such that none of us could do anything to take it down, except for pleading with the author to do so.

[UPDATE (Sep. 12th): It has been brought to my attention that we owe an apology not only for the most recent post, but also for at least some of the content of the other posts pertaining to the Oxford events, which had been posted a few months ago (now also deleted). So, for those too, our apologies. We are also looking into additional ways to make amends with the people negatively affected.]

The structure and moderation of the blog will change completely now; Jeffrey Ketland will no longer be a contributor (of his own initiative). The exact details still need to be discussed, but we hope to come back with something more concrete within a week or so.

Again, our apologies, to our readers and to those who were negatively affected by the post.

(And thank you Jeff, for all your otherwise very good work here at M-Phi over the years.)

UPDATE: the opinions of those who felt negatively affected by the posts are most welcome in comments below (or in private to me by email).

Tuesday, 9 September 2014

This is a short note just to say that I will not be contributing posts to M-Phi for the time being.

UPDATE: In view of recent events here at M-Phi, some important changes will take place regarding the management of the blog. We will talk more concretely about them in the near future, but for now let me say that we will do our utter best to restore the readers' trust in the blog, which may have been affected by recent developments.